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Adrian
I'm not sure exactly what your
function s does, but from what you describe, it just decides whether a string
can be mapped to the integers i a farily obvious way. While that gives you a
countably infinte set of symbols, that is not the same as the
integers.
If I remember correctly, you
need sixteen axioms to define the rational numbers, which include axioms such as
"One is not equal to zero" (or equivalent) and axioms for ordering numbers. The
integers differ from the rationals by not having a multiplicative inverse - and
therefore not having the axioms for multaplicative inverse. That is, the set of
axioms you choose defines the mathematical structures you generate, be they
grupoids, groups, rings, fields etc., and conversely, if you don't use all the
axioms for the integers, then the structures you generate include structures
which have different properties to those of the integers.
My understanding of Godel's
theorem is that once you can do the integers, you necessarily in a system which
contains undecidable propositions. This does not prevent you from having systems
which are decideable, but the "numbers" they contain will not actually be the
integers although they may have some properties which look the same.
Regards
Sean Barker
Bristol
----- Original Message -----
Sent: Saturday, December 12, 2009 6:54
PM
Subject: Re: [ontolog-forum] Form and
content
Sean,
Yes, if you let the full power of logic loose
above a description of the integers, the situation is undecidable.
At
the other extreme, you can just write simple grammar along
lines
int --> s(0) | s(int)
and you can use it to decide whether any input string is a positive
int. **
So my point is that a useful Logic (or other description
method) including the integers should be designed to be as expressive as
possible, while stopping short of
undecideability.
Cheers, -- Adrian
** And similarly for integers in everyday
decimal notation
Internet Business Logic
A Wiki and SOA Endpoint for
Executable Open Vocabulary English over SQL and RDF
Online at www.reengineeringllc.com
Shared use is free
Adrian Walker
Reengineering
On Sat, Dec 12, 2009 at 12:09 PM, sean barker <sean.barker@xxxxxxxxxxxxx>
wrote:
Adrian,
I believe the short answer
is no - it is a result of one of Godel's theorems. I guess
the starting point here is "Some Metamathematical Results on
Completeness and Consistency", 1930:
"If to the Peano axioms we add the logic of
Principia Mathematica (with the natural numbers as the individuals) together
with the axiom of choice (for all types) we obtain a formal system S for
which the following theorem's hold:
I : The system S is not complete: that is, it
contains propositions A for which neither A or NOT A is
provable..."
I don't know the area well enough to say at
which point non-completenss comes in, whether it is the axioms of Principia
Mathematica or the axiom of choice, or even just Peano's
axiomatisation.
Regards,
Sean Barker,
Bristol
-----
Original Message -----
Sent:
Thursday, December 10, 2009 9:46 PM
Subject:
Re: [ontolog-forum] Form and content
Hi Sean --
You wrote
...of course, you
wouldn't be able to axiomatise even the integers in many description
logics, since they are designed to prevent you describing problems that
are undecidable.
If that is
indeed so, it would seem to be shortcoming of those particular DLs.
There are
simple recursive definitions of the integers, and the question of whether
an arbitrary string satisfies such a definition is decidable.
Cheers, -- Adrian
Internet
Business Logic
A Wiki and SOA Endpoint for Executable Open
Vocabulary English over SQL and RDF
Online at www.reengineeringllc.com Shared use is
free
Adrian
Walker
Reengineering
On Thu, Dec 10, 2009 at 3:49 PM, sean barker <sean.barker@xxxxxxxxxxxxx> wrote:
Adrian,
The great thing about numbers is there are so many
different sorts to choose from: Naturals, integers, rationals, reals,
complex, quarternons and so on, not to mention the modulo groups, the
transfinite numbers and the floating point numbers (which, not
being a metric space, make a mess of most theorems about numbers). I
suspect if you get into the hard maths, then that will raise questions
about the axiom of choice and whether you should use it. And, of course,
you wouldn't be able to axiomatise even the integers in many description
logics, since they are designed to prevent you describing problems that
are undecidable.
Sean Barker
Bristol, UK
*** WARNING ***
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Keep this in mind if you answer this message.
Hi John, You wrote ...anybody who needs a
different ontology for numbers can use that instead.Somehow
I'm reminded of the saying "the nice thing about standards is that there
are so many of them to choose from" Seriously, if we can't agree
on a standard for something as basic as a number, what are the chances
for interoperability without the need for expensive and continuing human
intervention? To get a measure of how far there is still to go on
this journey, here's a real world problem description that serves as a
nice example of an interoperability requirement: http://www.economist.com/businessfinance/displaystory.cfm?story_id=15016132
Cheers, -- Adrian Internet Business Logic A Wiki and SOA
Endpoint for Executable Open Vocabulary English over SQL and
RDF Online at www.reengineeringllc.com Shared use
is free Adrian Walker Reengineering
Most people on this list understand the distinction
between form
and content, but some discussions tend to blur the
distinction.
I received an offline question related to that point,
and I thought
it might be useful to forward the answer to the
list.
John
Sowa
___________________________________________________________________
>
May I know what is the difference between semantic
network
> and ontology?
A semantic network is a
graphical form for knowledge representation.
It can be used to
express content of any kind.
An ontology is a formal definition
of content, which could be
represented in many different KR
languages and notations.
The distinction between form and
content is critical, but some
KR languages incorporate some
ontology into the basic notation.
Therefore, they would combine
some content with the form.
For example, a temporal logic is
likely to have at least a minimal
ontology for time built into the
notation and rules of inference.
A KR language that includes
arithmetic will have an ontology for
numbers and operations on
numbers built into the basic language.
Common Logic is a
version of logic that is as neutral as possible
about ontology.
For example, CL includes syntax for numerals, but
it does not
assume any axioms about the integers represented by
those numerals.
One application might represent integers of
arbitrary length,
but another might have an upper limit of 2^63.
Some people have
complained about the lack of an ontology for
numbers in CL.
But there are many different ontologies that can
be added to
CL as needed. One example is the Mathematical toolkit
in the
ISO standard for Z notation. It's a fine ontology,
it's
defined by an ISO standard, and anyone who wants it can use
it
in conjunction with CL. But anybody who needs a
different
ontology for numbers can use that instead.
John
Sowa
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